# Poisson distribution

Q1: A sample of n independent observations x1, x2, . . . , xn are obtained of a random

variable having a Poisson distribution with mean ?. Show that the maximum likelihood

estimate of ? is the sample mean [see the attached file]. Show that corresponding estimator (X) is an unbiased estimator of ?, and has variance ?/n.

The scientists Rutherford and Geiger reported an experiment in which they counted the

number of alpha particles emitted from a radioactive source during intervals of 7 1/2 seconds duration, for 2,612 different intervals. A total of 10,126 particles were counted. The data obtained are summarised in the table attached.

It has been suggested that the number of particles emitted in an interval may be adequately modelled by a Poisson distribution. Assuming this conjecture to be correct, find the

maximum likelihood estimate of the mean of this distribution, and use this to estimate the

expected frequencies corresponding to the observed frequencies given in the table. Comment

informally on the extent of agreement between these observed and expected frequencies.

Q2: A random sample X1, X2, . . . , Xn (n > 2) of independent observations is taken from

a random variable X which has a geometric distribution with parameter p (0 < p < 1),

whose probability function is given by [see the attached file].

Information was obtained on the birth order of children in 7745 families. The records

were examined to see when a family's first daughter was born. The birth order number for

the rst daughter and the data are given in the attachment.

It has been suggested that the birth order number of the first daughter may be adequately modelled by a geometric distribution. Assuming this conjecture to be correct, find

the maximum likelihood estimate of p (using the second result above) and estimate the

expected frequencies for the corresponding observed frequencies in the table. Comment

informally on the extent of agreement between these observed and expected frequencies.

Q3: A random sample X1, X2, . . . , Xn of independent observations is taken from a Gamma

distribution, whose density function is given by [see the attachment].

What reason would you give for supposing that the distribution is approximately normally distributed for large n?

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#### Solution Summary

Finding the maximum likelihood estimator from a Poisson distribution.

Statistics Problems solved in this posting are based on Poisson Distribution and Normal Distribution.

For Specific Problems, please see the posted problems.

Use the Poisson Distribution to find the indicated probability.

49)

For a certain type of fabric, the average number of defects in each square foot of fabric is 0.3. Find the probability that a randomly selected square foot of the fabric will contain more than one defect.

49)

______

A)

0.0369

B)

0.9631

C)

0.0333

D)

0.7778

50)

If the random variable x has a Poisson Distribution with mean find the probability that

50)

______

A)

0.33834

B)

0.13534

C)

0.73576

D)

0.27067

51)

A naturalist leads whale watch trips every morning in March. The number of whales seen has a Poisson distribution with a mean of 4.3. Find the probability that on a randomly selected trip, the number of whales seen is 3.

51)

______

A)

0.3596

B)

0.5394

C)

0.1798

D)

0.3057

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

52)

Define a standard normal distribution by identifying its shape and the numeric values for its mean and standard deviation. Mark the mean and the standard deviations on the curve. What do z scores measure? Relate the concept of z scores to the Empirical Rule.

52)

_____________

53)

Complete the following table for a distribution in which μ = 16. It might be helpful to make a diagram to help you determine the continuity factor for each entry.

53)

_____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost.

54)

More than 9 pounds

54)

______

A)

B)

C)

D)

55)

Between 9.5 pounds and 11 pounds

55)

______

A)

B)

C)

D)

If Z is a standard normal variable, find the probability.

56)

The probability that Z lies between 0 and 3.01

56)

______

A)

0.9987

B)

0.1217

C)

0.4987

D)

0.5013

57)

The probability that Z lies between -1.10 and -0.36

57)

______

A)

0.2237

B)

0.2239

C)

-0.2237

D)

0.4951

58)

The probability that Z is greater than -1.82

58)

______

A)

0.9656

B)

-0.0344

C)

0.0344

D)

0.4656

59)

P(-0.73 < Z < 2.27)

59)

______

A)

0.2211

B)

1.54

C)

0.7557

D)

0.4884

Assume that X has a normal distribution, and find the indicated probability.

60)

The mean is μ = 60.0 and the standard deviation is σ = 4.0.

Find the probability that X is less than 53.0.

60)

______

A)

0.0802

B)

0.9599

C)

0.5589

D)

0.0401

61)

The mean is μ = 137.0 and the standard deviation is σ = 5.3.

Find the probability that X is between 134.4 and 140.1.

61)

______

A)

1.0311

B)

0.6242

C)

0.4069

D)

0.8138

Find the indicated probability.

62)

The incomes of trainees at a local mill are normally distributed with a mean of $1100 and a standard deviation of $150. What percentage of trainees earn less than $900 a month?

62)

______

A)

35.31%

B)

9.18%

C)

90.82%

D)

40.82%

63)

The weekly salaries of teachers in one state are normally distributed with a mean of $490 and a standard deviation of $45. What is the probability that a randomly selected teacher earns more than $525 a week?

63)

______

A)

0.2177

B)

0.2823

C)

0.1003

D)

0.7823

64)

Assume that the weights of quarters are normally distributed with a mean of 5.67 g and a standard deviation 0.070 g. A vending machine will only accept coins weighing between 5.48 g and 5.82 g. What percentage of legal quarters will be rejected?

64)

______

A)

1.62%

B)

0.0196%

C)

2.48%

D)

1.96%

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

65)

A recent survey based on a random sample of n = 500 voters, predicted that the Independent candidate for the mayoral election will get 24% of the vote, but he actually gets 28%. Can it be concluded that the survey was done incorrectly?

65)

_____________

66)

A poll of 1200 randomly selected students in grades 6 through 8 was conducted and found that 39% enjoy playing sports. What is the sampling distribution suggested by the given data?

66)

_____________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Identify the probability of each sample, and describe the sampling distribution of the sample means.

67)

Personal phone calls received in the last three days by a new employee were 2, 6, and 8. Assume that samples of size 2 are randomly selected with replacement from this population of three values.

67)

______

A)

1/9; 2/9; 1/9; 0/9; 1/9; 0/9; 1/9; 2/9; 1/9

B)

1/9; 1/9; 1/9; 1/9; 1/9; 1/9; 1/9; 1/9; 1/9

C)

1/8; 1/9; 1/8; 1/9; 1/8; 1/9; 1/8; 1/9; 1/8

D)

1/3; 1/3; 1/18; 1/6; 1/18; 1/9; 1/6; 1/18; 1/9

For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation.

68)

n = 18 and p = .6

68)

______

A)

Normal approximation is suitable.

B)

Normal approximation is not suitable.

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.

69)

Estimate the probability of getting exactly 43 boys in 90 births.

69)

______

A)

0.0764

B)

0.0159

C)

0.0729

D)

0.1628

Use the normal distribution to approximate the desired probability.

70)

A coin is tossed 20 times. A person, who claims to have extrasensory perception, is asked to predict the outcome of each flip in advance. She predicts correctly on 11 tosses. What is the probability of being correct 11 or more times by guessing? Does this probability seem to verify her claim?

70)

______

A)

.4129 , yes

B)

.0871 , yes

C)

.4129 , no

D)

.0871 , no

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